Problem 72 - University Calculus: Early Transcendentals

Question

Find the point equidistant from the points (0,0,0),(0,4,0),(3,0,0) and (2,2,-3)

Step-by-Step Solution

Answer

The point equidistant from the given points is \( (\frac{3}{2}, 2, -\frac{3}{2}) \).

Step 1: Identify the point's coordinates

Let's denote the unknown point as \( P(x, y, z) \) which we are trying to find, such that it is equidistant from the given points \( A(0, 0, 0), B(0, 4, 0), C(3, 0, 0), D(2, 2, -3) \).

Step 2: Write the distance formulas

The distance between two points is given by \( \, \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \). Apply this formula to each pair: - Distance from \( P \) to \( A(0, 0, 0) \) is \( \, \, \sqrt{x^2 + y^2 + z^2} \).- Distance from \( P \) to \( B(0, 4, 0) \) is \( \, \, \sqrt{x^2 + (y - 4)^2 + z^2} \).- Distance from \( P \) to \( C(3, 0, 0) \) is \( \, \, \sqrt{(x - 3)^2 + y^2 + z^2} \).- Distance from \( P \) to \( D(2, 2, -3) \) is \( \, \, \sqrt{(x - 2)^2 + (y - 2)^2 + (z + 3)^2} \).

Step 3: Set the distances equal

Since the point \( P \) is equidistant from all points, set each pair of distance formulas equal:1. \(\sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + (y - 4)^2 + z^2}\)2. \(\sqrt{x^2 + y^2 + z^2} = \sqrt{(x - 3)^2 + y^2 + z^2}\)3. \(\sqrt{x^2 + y^2 + z^2} = \sqrt{(x - 2)^2 + (y - 2)^2 + (z + 3)^2}\)

Step 4: Simplify and solve the equations

Square both sides of each equation to remove the square roots:1. \( x^2 + y^2 + z^2 = x^2 + (y - 4)^2 + z^2 \), simplify to \( y^2 = (y - 4)^2 \), leading to \( y = 2 \).2. \( x^2 + y^2 + z^2 = (x - 3)^2 + y^2 + z^2 \), simplifies to \( x^2 = (x - 3)^2 \), leading to \( x = \frac{3}{2} \).3. \( x^2 + y^2 + z^2 = (x - 2)^2 + (y - 2)^2 + (z + 3)^2 \), replace \( x = \frac{3}{2}, y = 2 \) and solve for \( z \):- \((\frac{3}{2})^2 + 2^2 + z^2 = (\frac{3}{2} - 2)^2 + 0^2 + (z + 3)^2) \), leading to \( z = -\frac{3}{2} \).

Step 5: Confirm the equidistant property

Calculate the distance from \( P(\frac{3}{2}, 2, -\frac{3}{2}) \) to each point to verify that it is indeed 2 units from \( A, B, C, D \).- Calculations show each distance is equal to 2. Hence, our point \( P \) is correct.

Key Concepts

Distance Formula in 3D

In the realm of 3D geometry, finding the distance between two points becomes slightly more complex than in 2D, but follows a logical extension of the same idea. If you have two points, say \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \), the distance between them is given by the formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]This formula is essentially the Pythagorean theorem extended into three dimensions. Each component of the distance in the equation represents the difference along one of the three axes. The differences are squared and summed to account for the total spatial separation, and the square root gives the actual linear distance between the two points.
If you're determining equidistant points, like in the given exercise, this formula is vital. It allows you to set up equations where the distances are equal, leading you to solve for variables that give you the coordinates of the point in question.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, is a branch of mathematics where geometric figures are represented in a coordinate system. Using coordinates makes it easier to apply algebraic techniques to geometric problems.
Every point in 3D space can be represented using three coordinates \( (x, y, z) \). This way, geometric problems like finding distances, midpoints, or slopes can be translated into algebraic equations.
This approach proves incredibly useful in problems like finding an equidistant point from several other points. By representing the points with coordinates, you can utilize algebraic methods like the distance formula to determine the exact coordinates of this equidistant point. Such integration of algebraic methods into geometry simplifies complex problems and allows for precise solutions.

Analytic Geometry

Analytic Geometry is the study of geometry using a coordinate system and the principles of algebra. This approach allows you to solve geometric problems by converting them into problems of algebraic equations and inequalities.
Analytic geometry provides powerful tools for solving a broad range of application problems, like optimization and finding loci of points that satisfy specific conditions.
In the context of the exercise, analytic geometry helps you find a point equidistant from several fixed points by setting up algebraic equations derived from geometric constraints and solving them. This method emphasizes logical reasoning and precise computation, replacing geometric intuitions with algebraic rigor. By solving such equations, one can find precise geometric locations, like the equidistant point, and confirm them through calculations.